Let $(b_n)_{n \geq 1}$ be a decreasing sequence of positive real numbers such that $\sum_{n \geq 1} b_n = \infty$. And let $(a_n)_{n \geq 1}$ be another sequence of positive real numbers such that $\sum_{n \geq 1} a_nb_n =1$.
How to estimate $P_n := \prod_{i=1}^{n} a_i$?
I apreciate to conclude that $$\limsup P_n < 1.$$